# Super-Consistent Estimation of Points of Impact in Nonparametric Regression with Functional Predictors

## TL;DR

This paper introduces a super-consistent estimator for identifying specific impactful points in functional predictors within nonparametric regression models, improving accuracy without prior knowledge of model components.

## Contribution

The authors develop a novel estimator for points of impact that achieves super-consistent convergence and does not require pre-estimates of other model parts.

## Key findings

- Estimator has super-consistent convergence rate
- Method performs well in finite samples
- Applicable to nonparametric and generalized linear models

## Abstract

Predicting scalar outcomes using functional predictors is a classic problem in functional data analysis. In many applications, however, only specific locations or time-points of the functional predictors have an impact on the outcome. Such ``points of impact'' are typically unknown and have to be estimated in addition to estimating the usual model components. We show that our points of impact estimator enjoys a super-consistent convergence rate and does not require knowledge or pre-estimates of the unknown model components. This remarkable result facilitates the subsequent estimation of the remaining model components as shown in the theoretical part, where we consider the case of nonparametric models and the practically relevant case of generalized linear models. The finite sample properties of our estimators are assessed by means of a simulation study. Our methodology is motivated by data from a psychological experiment in which the participants were asked to continuously rate their emotional state while watching an affective video eliciting a varying intensity of emotional reactions.